Hamiltonian formalism of two-dimensional Vlasov kinetic equation

Abstract

In this paper, the two-dimensional Benney system describing long wave propagation of a finite depth fluid motion and the multi-dimensional RussoSmereka kinetic equation describing a bubbly flow are considered. The Hamiltonian approach established by J. Gibbons for the one-dimensional Vlasov kinetic equation is extended to a multi-dimensional case. A local Hamiltonian structure associated with the hydrodynamic lattice of moments derived by D. J. Benney is constructed. A relationship between this hydrodynamic lattice of moments and the twodimensional Vlasov kinetic equation is found. In the two-dimensional case, a Hamiltonian hydrodynamic lattice for the Russo-Smereka kinetic model is constructed. Simple hydrodynamic reductions are presented.

LÖwner evolution and finite-dimensional reductions of integrable systems

“…where their Hamiltonian densities are can be recalculated via moments A s . This means that the Benney hydrodynamic chain has at least two local Hamiltonian structures (see details in [14]). As in the previous particular case, we are looking for N components commuting the hydrodynamic reductions r i τ (ζ) = w i (r, ζ)r i x .…”

“…Infinitely many conservative laws (cf. (38)) (H −1 ) s + H 0 k = 0, 1, ... also can be written as the modified Benney hydrodynamic chain (see details in[14])whereH −1 = B 0 , H 0 = B 1 , H 1 = B 2 + B 0 B 1 − 1 12 (B 0 ) 3 ,... This Modified Benney hydrodynamic chain is related to the modified dKP equation (cf.…”